一类非线性抛物方程的初边值问题

The Initial and Boundary Value Problems to a Kind of Nonlinear Evolution Equations

本文研究了一类非线性抛物方程的初边值问题,即u_t-f(u)_(xx)=0,x∈R_+,f′(u)>0,u(0,t)=u_,t≥0;u(+∞,0)=u+.这里我们考虑一般情形,即u_≠u+.在某种小性条件下,我们证明了以上抛物方程的解存在且当时间充分大时,解趋近该问题的自相似解 u(x/1+t^(1/2) ).我们还进一步得到了解的最优衰减速度为(1+t)^(-1/4).

This paper studied the initial and boundary value problems to a kind of nonlinear parabolic equations：ut-f（u）xx = 0,x∈R＋,f＇（u）0,u（0,t） = u-,t≥0;u（＋∞,0） = u＋.The general case u-≠u＋ is considered.It is proved that under some smallness conditions the solutions of this problem tend to the self-similar solution u（x/1＋t1/2 ） as the time t goes to infinity.Furthermore,the optimal decay rate is also obtained which reads（1 ＋ t）-1/4.